This document outlines key concepts related to long-term investment opportunities and frictions. It discusses how investment opportunities depend on state variables that influence returns and risks over time. It also introduces the concepts of equivalent safe rate and equivalent annuity, which define the optimal growth rate of wealth or utility for a long-term investor. The document proposes solving for long-term optimal portfolios using duality bounds, stationary equations, and criteria for long-run optimality.
Hedging, Arbitrage, and Optimality with Superlinear Frictionsguasoni
In a continuous-time model with multiple assets described by cadlag processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. With such frictions, dual elements correspond to a pair of a shadow execution price combined with an equivalent martingale measure. For utility functions defined on the real line, optimal strategies exist even if arbitrage is present, because it is not scalable at will.
This presentation provides and overview of the paper "Jump-Diffusion Risk-Sensitive Asset Management." The paper proposes a solution to a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion ‘factor’ process.
Nonlinear Price Impact and Portfolio Choiceguasoni
In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. Trading rates are finite as with linear impact, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.
Hedging, Arbitrage, and Optimality with Superlinear Frictionsguasoni
In a continuous-time model with multiple assets described by cadlag processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. With such frictions, dual elements correspond to a pair of a shadow execution price combined with an equivalent martingale measure. For utility functions defined on the real line, optimal strategies exist even if arbitrage is present, because it is not scalable at will.
This presentation provides and overview of the paper "Jump-Diffusion Risk-Sensitive Asset Management." The paper proposes a solution to a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure,
with drifts that are functions of an auxiliary diffusion ‘factor’ process.
Nonlinear Price Impact and Portfolio Choiceguasoni
In a market with price-impact proportional to a power of the order flow, we derive optimal trading policies and their implied welfare for long-term investors with constant relative risk aversion, who trade one safe asset and one risky asset that follows geometric Brownian motion. These quantities admit asymptotic explicit formulas up to a structural constant that depends only on the price-impact exponent. Trading rates are finite as with linear impact, but they are lower near the target portfolio, and higher away from the target. The model nests the square-root impact law and, as extreme cases, linear impact and proportional transaction costs.
Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous...guasoni
We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua
Saddlepoint approximations, likelihood asymptotics, and approximate condition...jaredtobin
Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to the asset's volatility and liquidity. In a continuous-time model with one safe and one risky asset with constant investment opportunities and proportional transaction costs, we find the efficient portfolios that maximize long term expected returns for given average volatility. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines. For funds that seek to replicate multiples of index returns, such as leveraged ETFs, our efficient portfolios optimally trade off alpha against tracking error.
Asset Prices in Segmented and Integrated Marketsguasoni
This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.
Leveraged and inverse ETFs seek a daily return equal to a multiple of an index' return, an objective that requires continuous portfolio rebalancing. The resulting trading costs create a tradeoff between tracking error, which controls the short-term correlation with the index, and excess return (or tracking difference) -- the long-term deviation from the levered index' performance. With proportional trading costs, the optimal replication policy is robust to the index' dynamics. A summary of a fund's performance is the \emph{implied spread}, equal to the product of tracking error and excess return, rescaled for leverage and average volatility. The implies spread is insensitive to the benchmark's risk premium, and offers a tool to compare the performance of funds on the same benchmark, but with different multiples and tracking errors.
Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogeneous...guasoni
We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua
Saddlepoint approximations, likelihood asymptotics, and approximate condition...jaredtobin
Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.
When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to the asset's volatility and liquidity. In a continuous-time model with one safe and one risky asset with constant investment opportunities and proportional transaction costs, we find the efficient portfolios that maximize long term expected returns for given average volatility. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines. For funds that seek to replicate multiples of index returns, such as leveraged ETFs, our efficient portfolios optimally trade off alpha against tracking error.
Asset Prices in Segmented and Integrated Marketsguasoni
This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.
Leveraged and inverse ETFs seek a daily return equal to a multiple of an index' return, an objective that requires continuous portfolio rebalancing. The resulting trading costs create a tradeoff between tracking error, which controls the short-term correlation with the index, and excess return (or tracking difference) -- the long-term deviation from the levered index' performance. With proportional trading costs, the optimal replication policy is robust to the index' dynamics. A summary of a fund's performance is the \emph{implied spread}, equal to the product of tracking error and excess return, rescaled for leverage and average volatility. The implies spread is insensitive to the benchmark's risk premium, and offers a tool to compare the performance of funds on the same benchmark, but with different multiples and tracking errors.
Shortfall aversion reflects the higher utility loss of a spending cut from a reference point than the utility gain from a similar spending increase, in the spirit of Prospect Theory's loss aversion. This paper posits a model of utility of spending scaled by a function of past peak spending, called target spending. The discontinuity of the marginal utility at the target spending corresponds to shortfall aversion. According to the closed-form solution of the associated spending-investment problem, (i) the spending rate is constant and equals the historical peak for relatively large values of wealth/target; and (ii) the spending rate increases (and the target with it) when that ratio reaches its model-determined upper bound. These features contrast with traditional Merton-style models which call for spending rates proportional to wealth. A simulation using the 1926-2012 realized returns suggests that spending of the very shortfall averse is typically increasing and very smooth.
Never selling stocks is optimal for investors with a long horizon and a realistic range of preference and market parameters, if relative risk aversion, investment opportunities, proportional transaction costs, and dividend yields are constant. Such investors should buy stocks when their portfolio weight is too low, and otherwise hold them, letting dividends rebalance to cash over time rather than selling. With capital gain taxes, this policy outperforms both static buy-and-hold and dynamic rebalancing strategies that account for transaction costs. Selling stocks becomes optimal if either their target weight is low, or intermediate consumption is substantial.
We develop a new method to optimize portfolios of options in a market where European calls and puts are available with many exercise prices for each of several potentially correlated underlying assets. We identify the combination of asset-specific option payoffs that maximizes the Sharpe ratio of the overall portfolio: such payoffs are the unique solution to a system of integral equations, which reduce to a linear matrix equation under suitable representations of the underlying probabilities. Even when implied volatilities are all higher than historical volatilities, it can be optimal to sell options on some assets while buying options on others, as hedging demand outweighs demand for asset-specific returns.
New Mathematical Tools for the Financial SectorSSA KPI
AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 5.
More info at http://summerschool.ssa.org.ua
Should Commodity Investors Follow Commodities' Prices?guasoni
Most institutional investors gain access to commodities through diversified index funds, even though mean-reverting prices and low correlation among commodities returns indicate that two-fund separation does not hold for commodities. In contrast to demand for stocks and bonds, we find that, on average, demand for commodities is largely insensitive to risk aversion, with intertemporal hedging demand playing a major role for more risk averse investors. Comparing the optimal strategies of investors who observe only the index to those of investors who observe all commodities, we find that information on commodity prices leads to significant welfare gains, even if trading is confined to the index only.
Investing on behalf of a firm, a trader can feign personal skill by committing fraud that with high probability remains undetected and generates small gains, but that with low probability bankrupts the firm, offsetting ostensible gains. Honesty requires enough skin in the game: if two traders with isoelastic preferences operate in continuous-time and one of them is honest, the other is honest as long as the respective fraction of capital is above an endogenous fraud threshold that depends on the trader’s preferences and skill. If both traders can cheat, they reach a Nash equilibrium in which the fraud threshold of each of them is lower than if the other one were honest. More skill, higher risk aversion, longer horizons, and greater volatility all lead to honesty on a wider range of capital allocations between the traders.
Basic concepts and how to measure price volatility
Presented by Carlos Martins-Filho at the AGRODEP Workshop on Analytical Tools for Food Prices
and Price Volatility
June 6-7, 2011 • Dakar, Senegal
For more information on the workshop or to see the latest version of this presentation visit: http://www.agrodep.org/first-annual-workshop
Similar to UT Austin - Portugal Lectures on Portfolio Choice (20)
American student loans are fixed-rate debt contracts that may be repaid in full by a certain maturity. Alternatively, income-based schemes give borrowers the option to make payments proportional to their income above subsistence for a number of years, after which the remaining balance is forgiven but taxed as ordinary income. The repayment strategy that minimizes the present value of future payments takes two possible forms: For a small loan balance, it is optimal to make maximum payments until the loan is fully repaid, forgoing both income-based schemes and loan forgiveness. For a large balance, enrolling in income-based schemes is optimal either immediately or after a period of maximum payments. Overall, the benefits of income-based schemes are substantial for large loan balances but negligible for small loans.
Compared with existing payment systems, Bitcoin’s throughput is low. Designed to address Bitcoin’s scalability challenge, the Lightning Network (LN) is a protocol allowing two parties to secure bitcoin payments and escrow holdings between them. In a lightning channel, each party commits collateral towards future payments to the counterparty and payments are cryptographically secured updates of collaterals. The network of channels increases transaction speed and reduces blockchain congestion. This paper (i) identifies conditions for two parties to optimally establish a channel, (ii) finds explicit formulas for channel costs, (iii) obtains the optimal collaterals and savings entailed, and (iv) derives the implied reduction in congestion of the blockchain. Unidirectional channels costs grow with the square-root of payment rates, while symmetric bidirectional channels with their cubic root. Asymmetric bidirectional channels are akin to unidirectional when payment rates are significantly different, otherwise to symmetric bidirectional.
Reference Dependence: Endogenous Anchors and Life-Cycle Investingguasoni
In a complete market, we find optimal portfolios for an investor whose satisfaction stems from both a payoff's intrinsic utility and its comparison with a reference, as specified by Koszegi and Rabin. In the regular regime, arising when reference-dependence is low, the marginal utility of the optimal payoff is proportional to a twist of the pricing kernel. High reference-dependence leads to the anchors regime, whereby investors reduce disappointment by concentrating significant probability in one or few fixed outcomes, and multiple personal equilibria arise. If stocks follow geometric Brownian motion, the model implies that younger investors have larger stocks positions than older investors, highlighting the suggestion that reference-dependence helps explain this typical recommendation of financial planners.
A monopolist platform (the principal) shares profits with a population of affiliates (the agents), heterogeneous in skill, by offering them a common nonlinear contract contingent on individual revenue. The principal cannot discriminate across individual skill, but knows its distribution and aims at maximizing profits. This paper identifies the optimal contract, its implied profits, and agents' effort as the unique solution to an equation depending on skill distribution and agents' costs of effort. If skill is Pareto-distributed and agents' costs include linear and power components, closed-form solutions highlight two regimes: If linear costs are low, the principal's share of revenues is insensitive to skill distribution, and decreases as agents' costs increase. If linear costs are high, the principal's share is insensitive to the agents' costs and increases as inequality in skill increases.
Health-care slows the natural growth of mortality, indirectly increasing utility from consumption through longer lifetimes. This paper solves the problem of optimal dynamic consumption and healthcare spending with isoelastic utility, when natural mortality grows exponentially to reflect the Gompertz' law. Optimal consumption and healthcare imply an endogenous mortality law that is asymptotically exponential in the old-age limit, with lower growth rate than natural mortality. Health spending steadily increases with age, both in absolute terms and relative to total spending. Differential access to healthcare with isoelastic effects can account for observed longevity gains across cohorts.
how to sell pi coins on Bitmart crypto exchangeDOT TECH
Yes. Pi network coins can be exchanged but not on bitmart exchange. Because pi network is still in the enclosed mainnet. The only way pioneers are able to trade pi coins is by reselling the pi coins to pi verified merchants.
A verified merchant is someone who buys pi network coins and resell it to exchanges looking forward to hold till mainnet launch.
I will leave the telegram contact of my personal pi merchant to trade with.
@Pi_vendor_247
what is the best method to sell pi coins in 2024DOT TECH
The best way to sell your pi coins safely is trading with an exchange..but since pi is not launched in any exchange, and second option is through a VERIFIED pi merchant.
Who is a pi merchant?
A pi merchant is someone who buys pi coins from miners and pioneers and resell them to Investors looking forward to hold massive amounts before mainnet launch in 2026.
I will leave the telegram contact of my personal pi merchant to trade pi coins with.
@Pi_vendor_247
Even tho Pi network is not listed on any exchange yet.
Buying/Selling or investing in pi network coins is highly possible through the help of vendors. You can buy from vendors[ buy directly from the pi network miners and resell it]. I will leave the telegram contact of my personal vendor.
@Pi_vendor_247
What price will pi network be listed on exchangesDOT TECH
The rate at which pi will be listed is practically unknown. But due to speculations surrounding it the predicted rate is tends to be from 30$ — 50$.
So if you are interested in selling your pi network coins at a high rate tho. Or you can't wait till the mainnet launch in 2026. You can easily trade your pi coins with a merchant.
A merchant is someone who buys pi coins from miners and resell them to Investors looking forward to hold massive quantities till mainnet launch.
I will leave the telegram contact of my personal pi vendor to trade with.
@Pi_vendor_247
how can i use my minded pi coins I need some funds.DOT TECH
If you are interested in selling your pi coins, i have a verified pi merchant, who buys pi coins and resell them to exchanges looking forward to hold till mainnet launch.
Because the core team has announced that pi network will not be doing any pre-sale. The only way exchanges like huobi, bitmart and hotbit can get pi is by buying from miners.
Now a merchant stands in between these exchanges and the miners. As a link to make transactions smooth. Because right now in the enclosed mainnet you can't sell pi coins your self. You need the help of a merchant,
i will leave the telegram contact of my personal pi merchant below. 👇 I and my friends has traded more than 3000pi coins with him successfully.
@Pi_vendor_247
The European Unemployment Puzzle: implications from population agingGRAPE
We study the link between the evolving age structure of the working population and unemployment. We build a large new Keynesian OLG model with a realistic age structure, labor market frictions, sticky prices, and aggregate shocks. Once calibrated to the European economy, we quantify the extent to which demographic changes over the last three decades have contributed to the decline of the unemployment rate. Our findings yield important implications for the future evolution of unemployment given the anticipated further aging of the working population in Europe. We also quantify the implications for optimal monetary policy: lowering inflation volatility becomes less costly in terms of GDP and unemployment volatility, which hints that optimal monetary policy may be more hawkish in an aging society. Finally, our results also propose a partial reversal of the European-US unemployment puzzle due to the fact that the share of young workers is expected to remain robust in the US.
how can I sell pi coins after successfully completing KYCDOT TECH
Pi coins is not launched yet in any exchange 💱 this means it's not swappable, the current pi displaying on coin market cap is the iou version of pi. And you can learn all about that on my previous post.
RIGHT NOW THE ONLY WAY you can sell pi coins is through verified pi merchants. A pi merchant is someone who buys pi coins and resell them to exchanges and crypto whales. Looking forward to hold massive quantities of pi coins before the mainnet launch.
This is because pi network is not doing any pre-sale or ico offerings, the only way to get my coins is from buying from miners. So a merchant facilitates the transactions between the miners and these exchanges holding pi.
I and my friends has sold more than 6000 pi coins successfully with this method. I will be happy to share the contact of my personal pi merchant. The one i trade with, if you have your own merchant you can trade with them. For those who are new.
Message: @Pi_vendor_247 on telegram.
I wouldn't advise you selling all percentage of the pi coins. Leave at least a before so its a win win during open mainnet. Have a nice day pioneers ♥️
#kyc #mainnet #picoins #pi #sellpi #piwallet
#pinetwork
Introduction to Indian Financial System ()Avanish Goel
The financial system of a country is an important tool for economic development of the country, as it helps in creation of wealth by linking savings with investments.
It facilitates the flow of funds form the households (savers) to business firms (investors) to aid in wealth creation and development of both the parties
Currently pi network is not tradable on binance or any other exchange because we are still in the enclosed mainnet.
Right now the only way to sell pi coins is by trading with a verified merchant.
What is a pi merchant?
A pi merchant is someone verified by pi network team and allowed to barter pi coins for goods and services.
Since pi network is not doing any pre-sale The only way exchanges like binance/huobi or crypto whales can get pi is by buying from miners. And a merchant stands in between the exchanges and the miners.
I will leave the telegram contact of my personal pi merchant. I and my friends has traded more than 6000pi coins successfully
Tele-gram
@Pi_vendor_247
Financial Assets: Debit vs Equity Securities.pptxWrito-Finance
financial assets represent claim for future benefit or cash. Financial assets are formed by establishing contracts between participants. These financial assets are used for collection of huge amounts of money for business purposes.
Two major Types: Debt Securities and Equity Securities.
Debt Securities are Also known as fixed-income securities or instruments. The type of assets is formed by establishing contracts between investor and issuer of the asset.
• The first type of Debit securities is BONDS. Bonds are issued by corporations and government (both local and national government).
• The second important type of Debit security is NOTES. Apart from similarities associated with notes and bonds, notes have shorter term maturity.
• The 3rd important type of Debit security is TRESURY BILLS. These securities have short-term ranging from three months, six months, and one year. Issuer of such securities are governments.
• Above discussed debit securities are mostly issued by governments and corporations. CERTIFICATE OF DEPOSITS CDs are issued by Banks and Financial Institutions. Risk factor associated with CDs gets reduced when issued by reputable institutions or Banks.
Following are the risk attached with debt securities: Credit risk, interest rate risk and currency risk
There are no fixed maturity dates in such securities, and asset’s value is determined by company’s performance. There are two major types of equity securities: common stock and preferred stock.
Common Stock: These are simple equity securities and bear no complexities which the preferred stock bears. Holders of such securities or instrument have the voting rights when it comes to select the company’s board of director or the business decisions to be made.
Preferred Stock: Preferred stocks are sometime referred to as hybrid securities, because it contains elements of both debit security and equity security. Preferred stock confers ownership rights to security holder that is why it is equity instrument
<a href="https://www.writofinance.com/equity-securities-features-types-risk/" >Equity securities </a> as a whole is used for capital funding for companies. Companies have multiple expenses to cover. Potential growth of company is required in competitive market. So, these securities are used for capital generation, and then uses it for company’s growth.
Concluding remarks
Both are employed in business. Businesses are often established through debit securities, then what is the need for equity securities. Companies have to cover multiple expenses and expansion of business. They can also use equity instruments for repayment of debits. So, there are multiple uses for securities. As an investor, you need tools for analysis. Investment decisions are made by carefully analyzing the market. For better analysis of the stock market, investors often employ financial analysis of companies.
how to sell pi coins effectively (from 50 - 100k pi)DOT TECH
Anywhere in the world, including Africa, America, and Europe, you can sell Pi Network Coins online and receive cash through online payment options.
Pi has not yet been launched on any exchange because we are currently using the confined Mainnet. The planned launch date for Pi is June 28, 2026.
Reselling to investors who want to hold until the mainnet launch in 2026 is currently the sole way to sell.
Consequently, right now. All you need to do is select the right pi network provider.
Who is a pi merchant?
An individual who buys coins from miners on the pi network and resells them to investors hoping to hang onto them until the mainnet is launched is known as a pi merchant.
debuts.
I'll provide you the Telegram username
@Pi_vendor_247
Turin Startup Ecosystem 2024 - Ricerca sulle Startup e il Sistema dell'Innov...Quotidiano Piemontese
Turin Startup Ecosystem 2024
Una ricerca de il Club degli Investitori, in collaborazione con ToTeM Torino Tech Map e con il supporto della ESCP Business School e di Growth Capital
when will pi network coin be available on crypto exchange.DOT TECH
There is no set date for when Pi coins will enter the market.
However, the developers are working hard to get them released as soon as possible.
Once they are available, users will be able to exchange other cryptocurrencies for Pi coins on designated exchanges.
But for now the only way to sell your pi coins is through verified pi vendor.
Here is the telegram contact of my personal pi vendor
@Pi_vendor_247
USDA Loans in California: A Comprehensive Overview.pptxmarketing367770
USDA Loans in California: A Comprehensive Overview
If you're dreaming of owning a home in California's rural or suburban areas, a USDA loan might be the perfect solution. The U.S. Department of Agriculture (USDA) offers these loans to help low-to-moderate-income individuals and families achieve homeownership.
Key Features of USDA Loans:
Zero Down Payment: USDA loans require no down payment, making homeownership more accessible.
Competitive Interest Rates: These loans often come with lower interest rates compared to conventional loans.
Flexible Credit Requirements: USDA loans have more lenient credit score requirements, helping those with less-than-perfect credit.
Guaranteed Loan Program: The USDA guarantees a portion of the loan, reducing risk for lenders and expanding borrowing options.
Eligibility Criteria:
Location: The property must be located in a USDA-designated rural or suburban area. Many areas in California qualify.
Income Limits: Applicants must meet income guidelines, which vary by region and household size.
Primary Residence: The home must be used as the borrower's primary residence.
Application Process:
Find a USDA-Approved Lender: Not all lenders offer USDA loans, so it's essential to choose one approved by the USDA.
Pre-Qualification: Determine your eligibility and the amount you can borrow.
Property Search: Look for properties in eligible rural or suburban areas.
Loan Application: Submit your application, including financial and personal information.
Processing and Approval: The lender and USDA will review your application. If approved, you can proceed to closing.
USDA loans are an excellent option for those looking to buy a home in California's rural and suburban areas. With no down payment and flexible requirements, these loans make homeownership more attainable for many families. Explore your eligibility today and take the first step toward owning your dream home.
USDA Loans in California: A Comprehensive Overview.pptx
UT Austin - Portugal Lectures on Portfolio Choice
1. Long Run Investment: Opportunities and Frictions
Paolo Guasoni
Boston University and Dublin City University
July 2-6, 2012
CoLab Mathematics Summer School
2. Outline
• Long Run and Stochastic Investment Opportunities
• High-water Marks and Hedge Fund Fees
• Transaction Costs
• Price Impact
• Open Problems
3. Long Run and Stochastic Investment Opportunities
One
Long Run and Stochastic Investment Opportunities
Based on
Portfolios and Risk Premia for the Long Run
with Scott Robertson
4. Long Run and Stochastic Investment Opportunities
Independent Returns?
• Higher yields tend to be followed by higher long-term returns.
• Should not happen if returns independent!
5. Long Run and Stochastic Investment Opportunities
Utility Maximization
• Basic portfolio choice problem: maximize utility from terminal wealth:
π
max E[U(XT )]
π
• Easy for logarithmic utility U(x) = log x.
Myopic portfolio π = Σ−1 µ optimal. Numeraire argument.
• Portfolio does not depend on horizon (even random!), and on the
dynamics of the the state variable, but only its current value.
• But logarithmic utility leads to counterfactual predictions.
And implies that unhedgeable risk premia η are all zero.
• Power utility U(x) = x 1−γ /(1 − γ) is more flexible.
Portfolio no longer myopic. Risk premia η nonzero, and depend on γ.
• Power utility far less tractable.
Joint dependence on horizon and state variable dynamics.
• Explicit solutions few and cumbersome.
• Goal: keep dependence from state variable dynamics, lose from horizon.
• Tool: assume long horizon.
6. Long Run and Stochastic Investment Opportunities
Asset Prices and State Variables
t
• Safe asset St0 = exp 0
r (Ys )ds , d risky assets, k state variables.
dSti
=r (Yt )dt + dRti 1≤i ≤d
Sti
n
dRti =µi (Yt )dt + σij (Yt )dZtj 1≤i ≤d
j=1
k
dYti =bi (Yt )dt + aij (Yt )dWtj 1≤i ≤k
j=1
d Zi, Wj t =ρij (Yt )dt 1 ≤ i ≤ d, 1 ≤ j ≤ k
• Z , W Brownian Motions.
• Σ(y ) = (σσ )(y ), Υ(y ) = (σρa )(y ), A(y ) = (aa )(y ).
Assumption
r ∈ C γ (E, R), b ∈ C 1,γ (E, Rk ), µ ∈ C 1,γ (E, Rn ), A ∈ C 2,γ (E, Rk ×k ),
Σ ∈ C 2,γ (E, Rn×n ) and Υ ∈ C 2,γ (E, Rn×k ). The symmetric matrices A and Σ
¯
are strictly positive definite for all y ∈ E. Set Σ = Σ−1
7. Long Run and Stochastic Investment Opportunities
State Variables
• Investment opportunities:
safe rate r , excess returns µ, volatilities σ, and correlations ρ.
• State variables: anything on which investment opportunities depend.
• Example with predictable returns:
dRt =Yt dt + σdZt
dYt = − λYt dt + dWt
• State variable is expected return. Oscillates around zero.
• Example with stochastic volatility:
dRt =νYt dt + Yt dZt
dYt =κ(θ − Yt )dt + a Yt dWt
• State variable is squared volatility. Oscillates around positive value.
• State variables are generally stationary processes.
8. Long Run and Stochastic Investment Opportunities
(In)Completeness
• Υ Σ−1 Υ: covariance of hedgeable state shocks:
Measures degree of market completeness.
• A = Υ Σ−1 Υ: complete market.
State variables perfectly hedgeable, hence replicable.
• Υ = 0: fully incomplete market.
State shocks orthogonal to returns.
• Otherwise state variable partially hedgeable.
• One state: Υ Σ−1 Υ/a2 = ρ ρ.
Equivalent to R 2 of regression of state shocks on returns.
9. Long Run and Stochastic Investment Opportunities
Well Posedness
Assumption
There exists unique solution P (r ,y ) ) r ∈Rn ,y ∈E
to martingale problem:
n+k n+k
1 ˜ ∂2 ˜ ∂ ˜ Σ Υ ˜ µ
L= Ai,j (x) + bi (x) A= b=
2 ∂xi ∂xj ∂xi Υ A b
i,j=1 i=1
• Ω = C([0, ∞), Rn+k ) with uniform convergence on compacts.
• B Borel σ-algebra, (Bt )t≥0 natural filtration.
Definition
(P x )x∈Rn ×E on (Ω, B) solves martingale problem if, for all x ∈ Rn × E:
• P x (X0 = x) = 1
• P x (Xt ∈ Rn × E, ∀t ≥ 0) = 1
t
• f (Xt ) − f (X0 ) − 0
(Lf )(Xu )du is P x -martingale for all f ∈ C0 (Rn × E)
2
10. Long Run and Stochastic Investment Opportunities
Trading and Payoffs
Definition
Trading strategy: process (πti )1≤i≤d , adapted to Ft = Bt+ , the right-continuous
t≥0
envelope of the filtration generated by (R, Y ), and R-integrable.
• Investor trades without frictions. Wealth dynamics:
dXtπ
= r (Yt )dt + πt dRt
Xtπ
• In particular, Xtπ ≥ 0 a.s. for all t.
Admissibility implied by R-integrability.
11. Long Run and Stochastic Investment Opportunities
Equivalent Safe Rate
• Maximizing power utility E (XT )1−γ /(1 − γ) equivalent to maximizing the
π
1
certainty equivalent E (XT )1−γ
π 1−γ
.
• Observation: in most models of interest, wealth grows exponentially with
the horizon. And so does the certainty equivalent.
• Example: with r , µ, Σ constant, the certainty equivalent is exactly
1
exp (r + 2γ µ Σ−1 µ)T . Only total Sharpe ratio matters.
• Intuition: an investor with a long horizon should try to maximize the rate at
which the certainty equivalent grows:
1 1
β = max lim inf log E (XT )1−γ
π 1−γ
π T →∞ T
• Imagine a “dream” market, without risky assets, but only a safe rate ρ.
• If β < ρ, an investor with long enough horizon prefers the dream.
• If β > ρ, he prefers to wake up.
• At β = ρ, his dream comes true.
12. Long Run and Stochastic Investment Opportunities
Equivalent Annuity
• Exponential utility U(x) = −e−αx leads to a similar, but distinct idea.
• Suppose the safe rate is zero.
• Then optimal wealth typically grows linearly with the horizon, and so does
the certainty equivalent.
• Then it makes sense to consider the equivalent annuity:
1 π
β = max lim inf − log E e−αXT
π T →∞ αT
• The dream market now does not offer a higher safe rate, but instead a
stream of fixed payments, at rate ρ. The safe rate remains zero.
• The investor is indifferent between dream and reality for β = ρ.
• For positive safe rate, use definition with discounted quantities.
• Undiscounted equivalent annuity always infinite with positive safe rate.
13. Long Run and Stochastic Investment Opportunities
Solution Strategy
• Duality Bound.
• Stationary HJB equation and finite-horizon bounds.
• Criteria for long-run optimality.
14. Long Run and Stochastic Investment Opportunities
Stochastic Discount Factors
Definition
Stochastic discount factor: strictly positive adapted M = (Mt )t≥0 , such that:
y
EP Mt Sti Fs = Ms Ss
i
for all 0 ≤ s ≤ t, 0 ≤ i ≤ d
Martingale measure: probability Q, such that Q|Ft and P y |Ft equivalent for all
t ∈ [0, ∞), and discounted prices S i /S 0 Q-martingales for 1 ≤ i ≤ d.
• Martingale measures and stochastic discount factors related by:
dQ t
= exp 0 r (Ys )ds Mt
dP y Ft
• Local martingale property: all stochastic discount factors satisfy
t · ·
Mtη = exp − 0
rdt E − 0
(µ Σ−1 + η Υ Σ−1 )σdZ + 0
η adW t
for some adapted, Rk -valued process η.
• η represents the vector of unhedgeable risk premia.
• Intuitively, the Sharpe ratios of shocks orthogonal to dR.
15. Long Run and Stochastic Investment Opportunities
Duality Bound
π η
• For any payoff X = and any discount factor M = MT , E[XM] ≤ x.
XT
Because XM is a local martingale.
• Duality bound for power utility:
γ
1
1−γ
E X 1−γ 1−γ
≤ xE M 1−1/γ
• Proof: exercise with Hölder’s inequality.
• Duality bound for exponential utility:
1 x 1 M M
− log E e−αX ≤ + E log
α E[M] α E[M] E[M]
• Proof: Jensen inequality under risk-neutral densities.
• Both bounds true for any X and for any M.
Pass to sup over X and inf over M.
• Note how α disappears from the right-hand side.
• Both bounds in terms of certainty equivalents.
• As T → ∞, bounds for equivalent safe rate and annuity follow.
16. Long Run and Stochastic Investment Opportunities
Long Run Optimality
Definition (Power Utility)
An admissible portfolio π is long run optimal if it solves:
1 1
max lim inf log E (XT )1−γ 1−γ
π
π T →∞ T
The risk premia η are long run optimal if they solve:
γ
1 η 1−γ
min lim sup log E (MT )1−1/γ
η T →∞ T
Pair (π, η) long run optimal if both conditions hold, and limits coincide.
• Easier to show that (π, η) long run optimal together.
• Each η is an upper bound for all π and vice versa.
Definition (Exponential Utility)
Portfolio π and risk premia η long run optimal if they solve:
1 π η η
max lim inf − log E e−αXT 1
min lim sup T E MT log MT
π T →∞ T η T →∞
17. Long Run and Stochastic Investment Opportunities
HJB Equation
• V (t, x, y ) depends on time t, wealth x, and state variable y .
• Itô’s formula:
1
dV (t, Xt , Yt ) = Vt dt + Vx dXt + Vy dYt + (Vxx d X t + Vxy d X , Y t + Vyy d Y t )
2
• Vector notation. Vy , Vxy k -vectors. Vyy k × k matrix.
• Wealth dynamics:
dXt = (r + πt µt )Xt dt + Xt πt σt dZt
• Drift reduces to:
1 x2
Vt + xVx r + Vy b + tr(Vyy A) + xπ (µVx + ΥVxy ) + Vxx π Σπ
2 2
• Maximizing over π, the optimal value is:
Vx −1 Vxy −1
π=− Σ µ− Σ Υ
xVxx xVxx
• Second term is new. Interpretation?
18. Long Run and Stochastic Investment Opportunities
Intertemporal Hedging
V V
• π = − xVx Σ−1 µ − Σ−1 Υ xVxy
xx xx
• First term: optimal portfolio if state variable frozen at current value.
• Myopic solution, because state variable will change.
• Second term hedges shifts in state variables.
• If risk premia covary with Y , investors may want to use a portfolio which
covaries with Y to control its changes.
• But to reduce or increase such changes? Depends on preferences.
• When does second term vanish?
• Certainly if Υ = 0. Then no portfolio covaries with Y .
Even if you want to hedge, you cannot do it.
• Also if Vy = 0, like for constant r , µ and σ.
But then state variable is irrelevant.
• Any other cases?
19. Long Run and Stochastic Investment Opportunities
HJB Equation
• Maximize over π, recalling max(π b + 2 π Aπ = − 1 b A−1 b).
1
2
HJB equation becomes:
1 1 Σ−1
Vt + xVx r + Vy b + tr(Vyy A) − (µVx + ΥVxy ) (µVx + ΥVxy ) = 0
2 2 Vxx
• Nonlinear PDE in k + 2 dimensions. A nightmare even for k = 1.
• Need to reduce dimension.
• Power utility eliminates wealth x by homogeneity.
20. Long Run and Stochastic Investment Opportunities
Homogeneity
• For power utility, V (t, x, y ) = x 1−γ
1−γ v (t, y ).
x 1−γ
Vt = vt Vx = x −γ v Vxx = −γx −γ−1 v
1−γ
x 1−γ x 1−γ
Vxy = x −γ vy Vy = vy Vyy = vyy
1−γ 1−γ
• Optimal portfolio becomes:
1 −1 1 vy
π= Σ µ + Σ−1 Υ
γ γ v
• Plugging in, HJB equation becomes:
1 1−γ
vt + (1 − γ) r + µ Σ−1 µ v + b + Υ Σ−1 µ vy
2γ γ
1 1 − γ vy Υ Σ−1 Υvy
+ tr(vyy A) + =0
2 γ 2v
• Nonlinear PDE in k + 1 variables. Still hard to deal with.
21. Long Run and Stochastic Investment Opportunities
Long Run Asymptotics
• For a long horizon, use the guess v (t, y ) = e(1−γ)(β(T −t)+w(y )) .
• It will never satisfy the boundary condition. But will be close enough.
• Here β is the equivalent safe, to be found.
• We traded a function v (t, y ) for a function w(y ), plus a scalar β.
• The HJB equation becomes:
1 1−γ
−β + r + µ Σ−1 µ + b + Υ Σ−1 µ wy
2γ γ
1 1−γ 1−γ
+ tr(wyy A) + wy A− Υ Σ−1 Υ wy = 0
2 2 γ
• And the optimal portfolio:
1 −1 1
π= Σ µ+ 1− γ Σ−1 Υwy
γ
• Stationary portfolio. Depends on state variable, not horizon.
• HJB equation involved gradient wy and Hessian wyy , but not w.
• With one state, first-order ODE.
• Optimality? Accuracy? Boundary conditions?
22. Long Run and Stochastic Investment Opportunities
Example
• Stochastic volatility model:
dRt =νYt dt + Yt dZt
dYt =κ(θ − Yt )dt + ε Yt dWt
• Substitute values in stationary HJB equation:
ν2 1−γ
−β + r + y + κ(θ − y ) + ρενy wy
2γ γ
ε2 ε2 y 2 1−γ 2
+ wyy + (1 − γ) wy 1 − ρ =0
2 2 γ
• Try a linear guess w = λy . Set constant and linear terms to zero.
• System of equations in β and λ:
−β + r + κθλ =0
2
1−γ 2 1−γ ν2
λ2 (1 − γ) 1− ρ +λ νρ − κ + =0
2 γ γ 2γ
• Second equation quadratic, but only larger solution acceptable.
Need to pick largest possible β.
23. Long Run and Stochastic Investment Opportunities
Example (continued)
• Optimal portfolio is constant, but not the usual constant.
1
π= (ν + βρε)
γ
• Hedging component depends on various model parameters.
• Hedging is zero if ρ = 0 or ε = 0.
• ρ = 0: hedging impossible. Returns do not covary with state variable.
• ε = 0: hedging unnecessary. State variable deterministic.
• Hedging zero also if β = 0, which implies logarithmic utility.
• Logarithmic investor does not hedge, even if possible.
• Lives every day as if it were the last one.
• Equivalent safe rate:
θν 2 1 νρ
β= 1− 1− ε + O(ε2 )
2γ γ κ
• Correction term changes sign as γ crosses 1.
24. Long Run and Stochastic Investment Opportunities
Martingale Measure
• Many martingale measures. With incomplete market, local martingale
condition does not identify a single measure.
• For any arbitrary k -valued ηt , the process:
· ·
Mt = E − (µ Σ−1 + η Υ Σ−1 )σdZ + η adW
0 0 t
is a local martingale such that MR is also a local martingale.
π
• Recall that MT = yU (XT ).
• If local martingale M is a martingale, it defines a stochastic discount factor.
1 −1
• π= γΣ (µ + Υ(1 − γ)wy ) yields:
T T
µ− 1 π Σπ)dt−γ
U (XT ) = (XT )−γ =x −γ e−γ
π π 0
(π 2 0
π σdZt
T
+(1−γ)wy Υ )Σ−1 σdZt + T
=e− 0
(µ 0
(... )dt
• Mathing the two expressions, we guess η = (1 − γ)wy .
25. Long Run and Stochastic Investment Opportunities
Risk Neutral Dynamics
• To find dynamics of R and Y under Q, recall Girsanov Theorem.
• If Mt has previous representation, dynamics under Q is:
˜
dRt =σd Zt
˜
dYt =(b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)η)dt + ad Wt
• Since η = (1 − γ)wy , it follows that:
˜
dRt =σd Zt
˜
dYt = b − Υ Σ−1 µ + (A − Υ Σ−1 Υ)(1 − γ)wy dt + ad Wt
• Formula for (long-run) risk neutral measure for a given risk aversion.
• For γ = 1 (log utility) boils down to minimal martingale measure.
• Need to find w to obtain explicit solution.
• And need to check that above martingale problem has global solution.
26. Long Run and Stochastic Investment Opportunities
Exponential Utility
• Instead of homogeneity, recall that wealth factors out of value function.
• Long-run guess: V (x, y , t) = e−αx+αβt+w(y ) .
β is now equivalent annuity.
• Set r = 0, otherwise safe rate wipes out all other effects.
• The HJB equation becomes:
1 1 1
−β + µ Σ−1 µ + b − Υ Σ−1 µ wy + tr(wyy A)− wy A − Υ Σ−1 Υ wy = 0
2 2 2
• And the optimal portfolio:
1 −1
Σ µ − Σ−1 Υwy
xπ =
α
• Rule of thumb to obtain exponential HJB equation:
˜
write power HJB equation in terms of w = γw, then send γ ↑ ∞.
Then remove the˜
• Exponential utility like power utility with “∞” relative risk aversion.
• Risk-neutral dynamics is minimal entropy martingale measure:
˜
dYt = b − Υ Σ−1 µ − (A − Υ Σ−1 Υ)wy dt + ad Wt
27. Long Run and Stochastic Investment Opportunities
HJB Equation
Assumption
w ∈ C 2 (E, R) and β ∈ R solve the ergodic HJB equation:
1 ¯ 1−γ 1 ¯
r+ µ Σµ + w A − (1 −)Υ ΣΥ w+
2γ 2 γ
1 ¯ 1
w b − (1 − )Υ Σµ + tr AD 2 w = β
γ 2
• Solution must be guessed one way or another.
• PDE becomes ODE for a single state variable
• PDE becomes linear for logarithmic utility (γ = 1).
• Must find both w and β
28. Long Run and Stochastic Investment Opportunities
Myopic Probability
Assumption
ˆ
There exists unique solution (P r ,y )r ∈Rn ,y ∈Rk to to martingale problem
n+k n+k
ˆ 1 ˜ ∂2 ˆ ∂
L= Ai,j (x) + bi (x)
2 ∂xi ∂xj ∂xi
i,j=1 i=1
1
γ (µ + (1 − γ)Υ w)
ˆ
b=
b − (1 − 1 ¯ 1 ¯
Σµ + A − (1 − γ )Υ ΣΥ (1 − γ) w
γ )Υ
ˆ
• Under P, the diffusion has dynamics:
1 ˆ
dRt = γ (µ + (1 − γ)Υ w) dt + σd Zt
1 ¯ 1 ¯ ˆ
dYt = b − (1 − γ )Υ Σµ + (A − (1 − γ )Υ ΣΥ)(1 − γ) w dt + ad Wt
ˆ
• Same optimal portfolio as a logarthmic investor living under P.
29. Long Run and Stochastic Investment Opportunities
Finite horizon bounds
Theorem
Under previous assumptions:
1¯
π = Σ (µ + (1 − γ)Υ w) , η = (1 − γ) w
γ
satisfy the equalities:
1 1
y y 1−γ
EP (XT )1−γ
π 1−γ
= eβT +w(y ) EP e−(1−γ)w(YT )
ˆ
γ γ
γ−1 1−γ 1−γ 1−γ
y y
η
EP (MT ) γ = eβT +w(y ) EP e−
ˆ
γ w(YT )
• Bounds are almost the same. Differ in Lp norm
• Long run optimality if expectations grow less than exponentially.
30. Long Run and Stochastic Investment Opportunities
Path to Long Run solution
• Find candidate pair w, β that solves HJB equation.
• Different β lead to to different solutions w.
• Must find w corresponding to the lowest β that has a solution.
• Using w, check that myopic probabiity is well defined.
ˆ
Y does not explode under dynamics of P.
• Then finite horizon bounds hold.
• To obtain long run optimality, show that:
1 y 1−γ
lim sup log EP e− γ w(YT ) = 0
ˆ
T →∞ T
1 y
lim sup log EP e−(1−γ)w(YT ) = 0
ˆ
T →∞ T
31. Long Run and Stochastic Investment Opportunities
Proof of wealth bound (1)
ˆ
dP
dP |Ft , which equals to E(M), where:
• Z = ρW + ρB. Define Dt =
¯
t
Mt = ¯ ¯
−qΥ Σµ + A − qΥ ΣΥ v (a )−1 dWt
0
t
− ¯ ¯
q Σµ + ΣΥ v σ ρdBt
¯
0
• For the portfolio bound, it suffices to show that:
p
(XT ) = ep(βT +w(y )−w(YT )) DT
π
π 1
which is the same as log XT − p log DT = βT + w(y ) − w(YT ).
• The first term on the left-hand side is:
T T
π 1
log XT = r +π µ− π Σπ dt + π σdZt
0 2 0
32. Long Run and Stochastic Investment Opportunities
Proof of wealth bound (2)
• Set π = 1 ¯ π
1−p Σ (µ + pΥ w). log XT becomes:
T 1−2p ¯ p2 ¯ 1 p
2
¯
0
r+ 2(1−p) µ Σµ − (1−p)2
µ ΣΥ w − 2 (1−p)2 w Υ ΣΥ w dt
1 T ¯ 1 T ¯ ¯
+ 1−p 0
(µ + pΥ w) ΣσρdWt − 1−p 0
(µ + pΥ w) Σσ ρdBt
• Similarly, log DT /p becomes:
T p ¯ p ¯ p p(2−p) ¯
0
− 2(1−p)2 µ Σµ − (1−p)2
µ ΣΥ w − 2 w A+ (1−p)2
Υ ΣΥ w dt+
T 1 ¯ 1 T ¯ ¯
0
w a+ 1−p (µ + pΥ w) Σσρ dWt + 1−p 0
(µ + pΥ w) Σσ ρdBt
π
• Subtracting yields for log XT − log DT /p
T 1 ¯ p ¯ p p ¯
0
r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt
T
− 0
w adWt
33. Long Run and Stochastic Investment Opportunities
Proof of wealth bound (3)
• Now, Itô’s formula allows to substitute:
T T T
1
− w adWt = w(y ) − w(YT ) + w bdt + tr(AD 2 w)dt
0 0 2 0
• The resulting dt term matches the one in the HJB equation.
π
• log XT − log DT /p equals to βT + w(y ) − w(YT ).
34. Long Run and Stochastic Investment Opportunities
Proof of martingale bound (1)
• For the discount factor bound, it suffices to show that:
1 η 1 1
p−1 log MT − p log DT = 1−p (βT + w(y ) − w(YT ))
1 η
• The term p−1 log MT equals to:
1 T 1 ¯ p2 ¯
1−p 0
r + 2 µ Σµ + 2 w A − Υ ΣΥ w dt+
1 T ¯ 1 T ¯ ¯
p−1 0
p w a − (µ + pΥ w) Σσρ dWt + 1−p 0
(µ + pΥ w) Σσ ρdBt
1 1 η 1
• Subtracting p log DT yields for p−1 log MT − p log DT :
1 T 1 ¯ p ¯ p p ¯
1−p 0
r+ 2(1−p) µ Σµ + 1−p µ ΣΥ w + 2 w A+ 1−p Υ ΣΥ w dt
1 T
− 1−p 0
w adWt
35. Long Run and Stochastic Investment Opportunities
Proof of martingale bound (2)
T
• Replacing again 0
w adWt with Itô’s formula yields:
1 T 1 ¯ p ¯
1−p 0
(r + 2(1−p) µ Σµ + ( 1−p µ ΣΥ + b ) w+
1 p p ¯
2 tr(AD 2 w) + 2 w A+ 1−p Υ ΣΥ w)dt
1
+ 1−p (w(y ) − w(YT ))
• And the integral equals 1
1−p βT by the HJB equation.
36. Long Run and Stochastic Investment Opportunities
Exponential Utility
Theorem
If r = 0 and w solves equation:
1 ¯ 1 ¯ ¯ 1
µ Σµ − w A − Υ ΣΥ w+ w b − Υ Σµ + tr AD 2 w = β
2 2 2
and myopic dynamics is well posed:
ˆ
dRt =σd Zt
¯ ¯ ˆ
dYt = b − Υ Σµ − (A − Υ ΣΥ) w dt + ad Wt
Then for the portfolio and risk premia (π, η) given by:
1
xπ = Σ−1 µ − Σ−1 Υ w η=− w
α
finite-horizon bounds hold as:
1 y π 1 y
− log EP e−α(XT −x) =βT + log EP ew(y )−w(YT )
ˆ
α α
1 y η 1 y
EP [M log M η ] =βT + EP [w(y ) − w(YT )]
2 α ˆ
37. Long Run and Stochastic Investment Opportunities
Long-Run Optimality
Theorem
If, in addition to the assumptions for finite-horizon bounds,
ˆ
• the random variables (Yt )t≥0 are P y -tight in E for each y ∈ E;
• supy ∈E (1 − γ)F (y ) < +∞, where F ∈ C(E, R) is defined as:
µ Σ−1 µ (1−γ)2
r −β+
2γ − 2γ w Υ Σ−1 Υ w e−(1−γ)w γ>1
F = µ Σ−1 µ (1−γ)2 − 1−γ w
r −β+
2γ − 2 w A − Υ Σ−1 Υ w e γ γ<1
Then long-run optimality holds.
• Straightforward to check, once w is known.
• Tightness checked with some moment condition.
• Does not require transition kernel for Y under any probability.
38. Long Run and Stochastic Investment Opportunities
Proof of Long-Run Optimality (1)
• By the duality bound:
1 1 y η 1 1−p y
0 ≤ lim inf log EP (MT )q log EP [(XT )p ]
− π
T →∞ p T T
1 y η 1−p 1 y
≤ lim sup log EP (MT )q − lim inf log EP [(XT )p ]
π
T →∞ pT T →∞ pT
1−p y 1 1 y
= lim sup log EP e− 1−p v (YT ) − lim inf
ˆ log EP e−v (YT )
ˆ
T →∞ pT T →∞ pT
• For p < 0 enough to show lower bound
1 y 1
lim inf log EP exp − 1−p v (YT )
ˆ ≥0
T →∞ T
and upper bound:
1 y
lim supT →∞ T log EP [exp (−v (YT ))] ≤ 0
ˆ
• Lower bound follows from tightness.
39. Long Run and Stochastic Investment Opportunities
Proof of Long-Run Optimality (2)
• For upper bound, set:
1
Lf = f b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 f
• Then, for α ∈ R, the HJB equation implies that L (eαv ) equals to:
1
αeαv v b − qΥ Σ−1 µ + A − qΥ Σ−1 Υ v + 2 tr AD 2 v + 1 α v A v
2
= αeαv 1
2 v (1 + α)A − qΥ Σ−1 Υ v + pβ − pr + q µ Σ−1 µ
2
• Set α = −1, to obtain that:
q q
L e−v = e−v
v Υ Σ−1 Υ v − λ + pr − µ Σ−1 µ
2 2
• The boundedness hypothesis on F allows to conclude that:
y
EP e−v (YT ) ≤ e−v (y ) + (K ∨ 0) T
ˆ
whence
1 y
log EP e−v (YT ) ≤ 0
ˆ lim sup
T →∞ T
• 0 < p < 1 similar. Reverse inequalities for upper and lower bounds.
1
Use α = − 1−p for upper bound.
40. High-water Marks and Hedge Fund Fees
Two
High-water Marks and Hedge Fund Fees
Based on
The Incentives of Hedge Fund Fees and High-Water Marks
with Jan Obłoj
41. High-water Marks and Hedge Fund Fees
Two and Twenty
• Hedge Funds Managers receive two types of fees.
• Regular fees, like Mutual Funds.
• Unlike Mutual Funds, Performance Fees.
• Regular fees:
a fraction ϕ of assets under management. 2% typical.
• Performance fees:
a fraction α of trading profits. 20% typical.
• High-Water Marks:
Performance fees paid after losses recovered.
42. High-water Marks and Hedge Fund Fees
High-Water Marks
Time Gross Net High-Water Mark Fees
0 100 100 100 0
1 110 108 108 2
2 100 100 108 2
3 118 116 116 4
• Fund assets grow from 100 to 110.
The manager is paid 2, leaving 108 to the fund.
• Fund drops from 108 to 100.
No fees paid, nor past fees reimbursed.
• Fund recovers from 100 to 118.
Fees paid only on increase from 108 to 118.
Manager receives 2.
43. High-water Marks and Hedge Fund Fees
High-Water Marks
2.5
2.0
1.5
1.0
0.5
20 40 60 80 100
44. High-water Marks and Hedge Fund Fees
Risk Shifting?
• Manager shares investors’ profits, not losses.
Does manager take more risk to increase profits?
• Option Pricing Intuition:
Manager has a call option on the fund value.
Option value increases with volatility. More risk is better.
• Static, Complete Market Fallacy:
Manager has multiple call options.
• High-Water Mark: future strikes depend on past actions.
• Option unhedgeable: cannot short (your!) hedge fund.
45. High-water Marks and Hedge Fund Fees
Questions
• Portfolio:
Effect of fees and risk-aversion?
• Welfare:
Effect on investors and managers?
• High-Water Mark Contracts:
consistent with any investor’s objective?
46. High-water Marks and Hedge Fund Fees
Answers
• Goetzmann, Ingersoll and Ross (2003):
Risk-neutral value of management contract (future fees).
Exogenous portfolio and fund flows.
• High-Water Mark contract worth 10% to 20% of fund.
• Panageas and Westerfield (2009):
Exogenous risky and risk-free asset.
Optimal portfolio for a risk-neutral manager.
Fees cannot be invested in fund.
• Constant risky/risk-free ratio optimal.
Merton proportion does not depend on fee size.
Same solution for manager with Hindy-Huang utility.
47. High-water Marks and Hedge Fund Fees
This Model
• Manager with Power Utility and Long Horizon.
Exogenous risky and risk-free asset.
Fees cannot be invested in fund.
• Optimal Portfolio:
1 µ
π=
γ ∗ σ2
γ ∗ =(1 − α)γ + α
γ =Manager’s Risk Aversion
α =Performance Fee (e.g. 20%)
• Manager behaves as if owned fund, but were more myopic (γ ∗ weighted
average of γ and 1).
• Performance fees α matter. Regular fees ϕ don’t.
48. High-water Marks and Hedge Fund Fees
Three Problems, One Solution
• Power utility, long horizon. No regular fees.
1 Manager maximizes utility of performance fees.
Risk Aversion γ.
2 Investor maximizes utility of wealth. Pays no fees.
Risk Aversion γ ∗ = (1 − α)γ + α.
3 Investor maximizes utility of wealth. Pays no fees.
Risk Aversion γ. Maximum Drawdown 1 − α.
• Same optimal portfolio:
1 µ
π=
γ ∗ σ2
49. High-water Marks and Hedge Fund Fees
Price Dynamics
dSt
= (r + µ)dt + σdWt (Risky Asset)
St
dSt α ∗
dXt = (r − ϕ)Xt dt + Xt πt St − rdt − 1−α dXt (Fund)
α
dFt = rFt dt + ϕXt dt + dX ∗ (Fees)
1−α t
Xt∗ = max Xs (High-Water Mark)
0≤s≤t
• One safe and one risky asset.
• Gain split into α for the manager and 1 − α for the fund.
• Performance fee is α/(1 − α) of fund increase.
50. High-water Marks and Hedge Fund Fees
Dynamics Well Posed?
• Problem: fund value implicit.
Find solution Xt for
dSt α
dXt = Xt πt − ϕXt dt − dX ∗
St 1−α t
• Yes. Pathwise construction.
Proposition
∗
The unique solution is Xt = eRt −αRt , where:
t t
σ2 2
Rt = µπs − π − ϕ ds + σ πs dWs
0 2 s 0
is the cumulative log return.
51. High-water Marks and Hedge Fund Fees
Fund Value Explicit
Lemma
Let Y be a continuous process, and α > 0.
Then Yt + 1−α Yt∗ = Rt if and only if Yt = Rt − αRt∗ .
α
Proof.
Follows from:
α α 1
Rt∗ = sup Ys + sup Yu = Yt∗ + Yt∗ = Y∗
s≤t 1 − α u≤s 1−α 1−α t
• Apply Lemma to cumulative log return.
52. High-water Marks and Hedge Fund Fees
Long Horizon
• The manager chooses the portfolio π which maximizes expected power
utility from fees at a long horizon.
• Maximizes the long-run objective:
1 p
max lim log E[FT ] = β
π T →∞ pT
• Dumas and Luciano (1991), Grossman and Vila (1992), Grossman and
Zhou (1993), Cvitanic and Karatzas (1995). Risk-Sensitive Control:
Bielecki and Pliska (1999) and many others.
2
1 µ
• β=r+ γ 2σ 2 for Merton problem with risk-aversion γ = 1 − p.
53. High-water Marks and Hedge Fund Fees
Solving It
• Set r = 0 and ϕ = 0 to simplify notation.
• Cumulative fees are a fraction of the increase in the fund:
α
Ft = (X ∗ − X0 )
∗
1−α t
• Thus, the manager’s objective is equivalent to:
1 ∗
max lim log E[(XT )p ]
π T →∞ pT
• Finite-horizon value function:
1
V (x, z, t) = sup E[XT p |Xt = x, Xt∗ = z]
∗
π p
1
dV (Xt , Xt∗ , t) = Vt dt + Vx dXt + Vxx d X t + Vz dXt∗
2
2
= Vt dt + Vz − α
1−α Vx dXt∗ + Vx Xt (πt µ − ϕ)dt + Vxx σ πt2 Xt2 dt
2
54. High-water Marks and Hedge Fund Fees
Dynamic Programming
• Hamilton-Jacobi-Bellman equation:
2
Vt + supπ xVx (πµ − ϕ) + Vxx σ π 2 x 2 ) x <z
2
α
Vz = 1−α Vx x =z
p
V = z /p
x =0
V = z p /p t =T
• Maximize in π, and use homogeneity
V (x, z, t) = z p /pV (x/z, 1, t) = z p /pu(x/z, 1, t).
2
ut − ϕxux − µ22 ux = 0 x ∈ (0, 1)
2σ uxx
ux (1, t) = p(1 − α)u(1, t) t ∈ (0, T )
u(x, T ) = 1
x ∈ (0, 1)
u(0, t) = 1 t ∈ (0, T )
55. High-water Marks and Hedge Fund Fees
Long-Run Heuristics
• Long-run limit.
Guess a solution of the form u(t, x) = ce−pβt w(x), forgetting the terminal
condition: 2
µ2 wx
−pβw − ϕxwx − 2σ2 wxx = 0 for x < 1
wx (1) = p(1 − α)w(1)
• This equation is time-homogeneous, but β is unknown.
• Any β with a solution w is an upper bound on the rate λ.
• Candidate long-run value function:
the solution w with the lowest β.
1−α µ2
• w(x) = x p(1−α) , for β = (1−α)γ+α 2σ 2 − ϕ(1 − α).
56. High-water Marks and Hedge Fund Fees
Verification
Theorem
µ2 1 1
If ϕ − r < 2σ 2
min γ∗ , γ∗2 , then for any portfolio π:
1 π p 1 µ2
lim log E (FT ) ≤ max (1 − α) + r − ϕ ,r
T ↑∞ pT γ ∗ 2σ 2
2
α 1 µ
Under the nondegeneracy condition ϕ + 1−α r < γ∗ 2σ 2 , the unique optimal
µ
solution is π = γ1∗ σ2 .
ˆ
• Martingale argument. No HJB equation needed.
• Show upper bound for any portfolio π (delicate).
• Check equality for guessed solution (easy).
57. High-water Marks and Hedge Fund Fees
Upper Bound (1)
• Take p > 0 (p < 0 symmetric).
• For any portfolio π:
T σ2 2 T ˜
RT = − π dt
0 2 t
+ 0
σπt d Wt
˜
• Wt = Wt + µ/σt risk-neutral Brownian Motion
• Explicit representation:
∗ ∗ µ ˜ µ2
E[(XT )p ] = E[ep(1−α)RT ] = EQ ep(1−α)RT e σ WT − 2σ2 T
π
• For δ > 1, Hölder’s inequality:
δ−1
δ µ ˜ µ2 δ
µ2
σ WT − 2σ 2
µ ˜ 1 T
∗ ∗
σ WT − 2σ 2 T
p(1−α)RT δp(1−α)RT δ δ−1
EQ e e ≤ EQ e EQ e
1 µ2
• Second term exponential normal moment. Just e δ−1 2σ2 T .
58. High-water Marks and Hedge Fund Fees
Upper Bound (2)
∗
• Estimate EQ eδp(1−α)RT .
• Mt = eRt strictly positive continuous local martingale.
Converges to zero as t ↑ ∞.
• Fact:
inverse of lifetime supremum (M∞ )−1 uniform on [0, 1].
∗
• Thus, for δp(1 − α) < 1:
∗ ∗ 1
EQ eδp(1−α)RT ≤ EQ eδp(1−α)R∞ =
1 − δp(1 − α)
• In summary, for 1 < δ < 1
p(1−α) :
1 π p 1 µ2
lim log E (FT ) ≤
T ↑∞ pT p(δ − 1) 2σ 2
• Thesis follows as δ → 1
p(1−α) .
59. High-water Marks and Hedge Fund Fees
High-Water Marks and Drawdowns
• Imagine fund’s assets Xt and manager’s fees Ft in the same account
Ct = Xt + Ft .
dSt
dCt = (Ct − Ft )πt
St
• Fees Ft proportional to high-water mark Xt∗ :
α
Ft = (X ∗ − X0 )
∗
1−α t
• Account increase dCt∗ as fund increase plus fees increase:
t t
α 1
Ct∗ − C0 =
∗ ∗
(dXs + dFs ) = ∗
+ 1 dXs = (X ∗ − X0 )
0 0 1−α 1−α t
• Obvious bound Ct ≥ Ft yields:
Ct ≥ α(Ct∗ − X0 )
• X0 negligible as t ↑ ∞. Approximate drawdown constraint.
Ct ≥ αCt∗
60. Transaction Costs
Three
Transaction Costs
Based on
Transaction Costs, Trading Volume, and the Liquidity Premium
with Stefan Gerhold, Johannes Muhle-Karbe, and Walter
Schachermayer
61. Transaction Costs
Model
• Safe rate r .
• Ask (buying) price of risky asset:
dSt
= (r + µ)dt + σdWt
St
• Bid price (1 − ε)St . ε is the spread.
• Investor with power utility U(x) = x 1−γ /(1 − γ).
• Maximize equivalent safe rate:
1
1 1−γ 1−γ
max lim log E XT
π T →∞ T
62. Transaction Costs
Wealth Dynamics
• Number of shares must have a.s. locally finite variation.
• Otherwise infinite costs in finite time.
• Strategy: predictable process (ϕ0 , ϕ) of finite variation.
• ϕ0 units of safe asset. ϕt shares of risky asset at time t.
t
• ϕt = ϕ↑ − ϕ↓ . Shares bought ϕ↑ minus shares sold ϕ↓ .
t t t t
• Self-financing condition:
St St
dϕ0 = −
t dϕ↑ + (1 − ε) 0 dϕ↓
t
St0 St
• Xt0 = ϕ0 St0 , Xt = ϕt St safe and risky wealth, at ask price St .
t
dXt0 =rXt0 dt − St dϕ↑ + (1 − ε)St dϕ↓ ,
t t
dXt =(µ + r )Xt dt + σXt dWt + St dϕ↑ − St dϕ↓
t
63. Transaction Costs
Control Argument
• V (t, x, y ) value function. Depends on time, and on asset positions.
• By Itô’s formula:
1
dV (t, Xt0 , Xt ) = Vt dt + Vx dXt0 + Vy dXt + Vyy d X , X t
2
σ2 2
= Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy dt
2 t
+ St (Vy − Vx )dϕ↑ + St ((1 − ε)Vx − Vy )dϕ↓ + σXt dWt
t t
• V (t, Xt0 , Xt ) supermartingale for any ϕ.
• ϕ↑ , ϕ↓ increasing, hence Vy − Vx ≤ 0 and (1 − ε)Vx − Vy ≤ 0
Vx 1
1≤ ≤
Vy 1−ε
64. Transaction Costs
No Trade Region
Vx 1
• When 1 ≤ Vy ≤ 1−ε does not bind, drift is zero:
σ2 2 Vx 1
Vt + rXt0 Vx + (µ + r )Xt Vy + X Vyy = 0 if 1 < < .
2 t Vy 1−ε
• This is the no-trade region.
• Long-run guess:
V (t, Xt0 , Xt ) = (Xt0 )1−γ v (Xt /Xt0 )e−(1−γ)(β+r )t
(1−γ)v (z) 1
• Set z = y /x. For 1 + z < v (z) < 1−ε + z, HJB equation is
2
σ 2
z v (z) + µzv (z) − (1 − γ)βv (z) = 0
2
• Linear second order ODE. But β unknown.
65. Transaction Costs
Smooth Pasting
(1−γ)v (z) 1
• Suppose 1 + z < v (z) < 1−ε + z same as l ≤ z ≤ u.
• For l < u to be found. Free boundary problem:
σ2 2
z v (z) + µzv (z) − (1 − γ)βv (z) = 0 if l < z < u,
2
(1 + l)v (l) − (1 − γ)v (l) = 0,
(1/(1 − ε) + u)v (u) − (1 − γ)v (u) = 0.
• Conditions not enough to find solution. Matched for any l, u.
• Smooth pasting conditions.
• Differentiate boundary conditions with respect to l and u:
(1 + l)v (l) + γv (l) = 0,
(1/(1 − ε) + u)v (u) + γv (u) = 0.
66. Transaction Costs
Solution Procedure
• Unknown: trading boundaries l, u and rate β.
• Strategy: find l, u in terms of β.
• Free boundary problem becomes fixed boundary problem.
• Find unique β that solves this problem.
67. Transaction Costs
Trading Boundaries
• Plug smooth-pasting into boundary, and result into ODE. Obtain:
2 2
l l
− σ (1 − γ)γ (1+l)2 v + µ(1 − γ) 1+l v − (1 − γ)βv = 0.
2
• Setting π− = l/(1 + l), and factoring out (1 − γ)v :
γσ 2 2
− π + µπ− − β = 0.
2 −
• π− risky weight on buy boundary, using ask price.
• Same argument for u. Other solution to quadratic equation is:
u(1−ε)
π+ = 1+u(1−ε) ,
• π+ risky weight on sell boundary, using bid price.
68. Transaction Costs
Gap
• Optimal policy: buy when “ask" weight falls below π− , sell when “bid"
weight rises above π+ . Do nothing in between.
• π− and π+ solve same quadratic equation. Related to β via
µ µ2 − 2βγσ 2
π± = ± .
γσ 2 γσ 2
• Set β = (µ2 − λ2 )/2γσ 2 . β = µ2 /2γσ 2 without transaction costs.
• Investor indifferent between trading with transaction costs asset with
volatility σ and excess return µ, and...
• ...trading hypothetical frictionless asset, with excess return µ2 − λ2 and
same volatility σ.
• µ− µ2 − λ2 is liquidity premium.
µ±λ
• With this notation, buy and sell boundaries are π± = γσ 2
.
69. Transaction Costs
Symmetric Trading Boundaries
• Trading boundaries symmetric around frictionless weight µ/γσ 2 .
• Each boundary corresponds to classical solution, in which expected
return is increased or decreased by the gap λ.
• With l(λ), u(λ) identified by π± in terms of λ, it remains to find λ.
• This is where the trouble is.
70. Transaction Costs
First Order ODE
• Use substitution:
log(z/l(λ))
w(y )dy l(λ)ey v (l(λ)ey )
v (z) = e(1−γ) , i.e. w(y ) = (1−γ)v (l(λ)ey )
• Then linear second order ODE becomes first order Riccati ODE
2µ µ−λ µ+λ
w (x) + (1 − γ)w(x)2 + σ2
− 1 w(x) − γ γσ 2 γσ 2
=0
µ−λ
w(0) =
γσ 2
µ+λ
w(log(u(λ)/l(λ))) =
γσ 2
2
u(λ) 1 π+ (1−π− ) 1 (µ+λ)(µ−λ−γσ )
where l(λ) = 1−ε π− (1−π+ ) = 1−ε (µ−λ)(µ+λ−γσ 2 ) .
• For each λ, initial value problem has solution w(λ, ·).
µ+λ
• λ identified by second boundary w(λ, log(u(λ)/l(λ))) = γσ 2
.
72. Transaction Costs
Shadow Market
• Find shadow price to make argument rigorous.
˜ ˜
• Hypothetical price S of frictionless risky asset, such that trading in S
withut transaction costs is equivalent to trading in S with transaction costs.
For optimal policy.
• For all other policies, shadow market is better.
• Use frictionless theory to show that candidate optimal policy is optimal in
shadow market.
• Then it is optimal also in transaction costs market.
73. Transaction Costs
Shadow Price as Marginal Rate of Substitution
˜
• Imagine trading is now frictionless, but price is St .
• Intuition: the value function must not increase by trading δ ∈ R shares.
• In value function, risky position valued at ask price St . Thus
˜
V (t, Xt0 − δ St , Xt + δSt ) ≤ V (t, Xt0 , Xt )
˜
• Expanding for small δ, −δVx St + δVy St ≤ 0
˜
• Must hold for δ positive and negative. Guess for St :
˜ Vy
St = St
Vx
log(Xt /lXt0 )
• Plugging guess V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t (Xt0 )1−γ e(1−γ) 0
w(y )dy
,
˜ w(Yt )
St = St
leYt (1
− w(Yt ))
• Shadow portfolio weight is precisely w:
w(Yt )
˜
ϕt St 1−w(Yt )
πt =
˜ = = w(Yt ),
˜
ϕ0 St0 + ϕt St 1 w(Yt )
+ 1−w(Yt )
t
74. Transaction Costs
Shadow Value Function
˜ ˜
• In terms of shadow wealth Xt = ϕ0 St0 + ϕt St , the value function becomes:
t
1−γ
Xt0 Yt
V (t, Xt , Yt ) = V (t, Xt0 , Xt ) = e−(1−γ)(r +β)t Xt1−γ
˜ ˜ ˜ e(1−γ) 0
w(y )dy
.
˜
Xt
˜ ˜
• Note that Xt0 /Xt = 1 − w(Yt ) by definition of St , and rewrite as:
1−γ
Xt0 Yt
w(y )dy Yt
e(1−γ) 0 = exp (1 − γ) log(1 − w(Yt )) + 0
w(y )dy
˜
Xt
Yt w (y )
= (1 − w(0))γ−1 exp (1 − γ) 0
w(y ) − 1−w(y ) dy .
˜
• Set w = w − w
1−w to obtain
Yt
V (t, Xt , Yt ) = e−(1−γ)(r +β)t Xt1−γ e(1−γ)
˜ ˜ ˜ 0
˜
w(y )dy
(1 − w(0))γ−1 .
• We are now ready for verification.
75. Transaction Costs
Construction of Shadow Price
• So far, a string of guesses. Now construction.
• Define Yt as reflected diffusion on [0, log(u/l)]:
dYt = (µ − σ 2 /2)dt + σdWt + dLt − dUt , Y0 ∈ [0, log(u/l)],
Lemma
˜ w(Y )
The dynamics of S = S leY (1−w(Y )) is given by
˜ ˜
d S(Yt )/S(Yt ) = (˜(Yt ) + r ) dt + σ (Yt )dWt ,
µ ˜
where µ(·) and σ (·) are defined as
˜ ˜
σ 2 w (y ) w (y ) σw (y )
µ(y ) =
˜ − (1 − γ)w(y ) , σ (y ) =
˜ .
w(y )(1 − w(y )) 1 − w(y ) w(y )(1 − w(y ))
˜
And the process S takes values within the bid-ask spread [(1 − ε)S, S].
76. Transaction Costs
Finite-horizon Bound
Theorem
˜
The shadow payoff XT of the portfolio π = w(Yt ) and the shadow discount
˜
· µ
˜ y
˜
factor MT = E(− 0 σ dWt )T satisfy (with q (y ) =
˜
˜
w(z)dz):
˜ 1−γ =e(1−γ)βT E e(1−γ)(q (Y0 )−q (YT )) ,
E XT ˆ ˜ ˜
1
γ γ
1− γ 1 ˜ ˜
E MT =e(1−γ)βT E e( γ −1)(q (Y0 )−q (YT ))
ˆ .
ˆ ˆ
where E[·] is the expectation with respect to the myopic probability P:
ˆ T T 2
dP µ
˜ 1 µ
˜
= exp − + σ π dWt −
˜˜ − + σπ
˜˜ dt .
dP 0 σ
˜ 2 0 σ
˜
77. Transaction Costs
First Bound (1)
˜ ˜ ˜ ˜
• µ, σ , π , w functions of Yt . Argument omitted for brevity.
˜
• For first bound, write shadow wealth X as:
˜ 1−γ = exp (1 − γ) T σ2 2
˜ T
XT 0
µπ −
˜˜ 2 π
˜ dt + (1 − γ) 0
σ π dWt .
˜˜
• Hence:
ˆ T 2
˜ 1−γ = d P exp σ2 2
˜ 1 µ
˜
XT dP 0
(1 − γ) µπ −
˜˜ 2 π
˜ + 2 − σ + σπ
˜ ˜˜ dt
T µ
˜
× exp 0
(1 − γ)˜ π − − σ + σ π
σ˜ ˜ ˜˜ dWt .
1 µ
˜
• Plug π =
˜ γ σ2
˜
+ (1 − γ) σ w . Second integrand is −(1 − γ)σ w.
σ
˜
˜ ˜
1µ˜2 2
• First integrand is 2 σ2
˜
+ γ σ π 2 − γ µπ , which equals to
˜
2 ˜ ˜˜
2 2
1−γ µ
˜
(1 − γ)2 σ w 2 +
2
˜ 2γ σ
˜
˜
+ σ(1 − γ)w .
78. Transaction Costs
First Bound (2)
1−γ
˜
• In summary, XT equals to:
ˆ T 2 2
dP 1 µ
˜
dP exp (1 − γ) 0
(1 − γ) σ w 2 +
2
˜ 2γ σ
˜
˜
+ σ(1 − γ)w dt
T
× exp −(1 − γ) 0
˜
σ wdWt .
˜ ˜
• By Itô’s formula, and boundary conditions w(0) = w(log(u/l)) = 0,
T 1 T
˜ ˜
q (YT ) − q (Y0 ) = 0
˜
w(Yt )dYt + 2 0
˜
w (Yt )d Y , Y t ˜ ˜
+ w(0)LT − w(u/l)UT
T σ2 2
σ T
= 0
µ− 2
˜
w+ 2
˜
w dt + 0
˜
σ wdWt .
T ˜ 1−γ equals to:
• Use identity to replace 0
˜
σ wdWt , and XT
ˆ
dP T
dP exp (1 − γ) 0
˜ ˜
(β) dt) × exp (−(1 − γ)(q (YT ) − q (Y0 ))) .
2
σ2 ˜ 2
σ2 1 µ
˜
as 2 w + (1 − γ) σ w 2 + µ −
2
˜ 2
˜
w+ 2γ σ
˜
˜
+ σ(1 − γ)w = β.
• First bound follows.
80. Transaction Costs
Buy and Sell at Boundaries
Lemma
˜ ˜
For 0 < µ/γσ 2 = 1, the number of shares ϕt = w(Yt )Xt /St follows:
dϕt µ−λ µ+λ
= 1− dLt − 1 − dUt .
ϕt γσ 2 γσ 2
˜
Thus, ϕt increases only when Yt = 0, that is, when St equals the ask price,
˜
and decreases only when Yt = log(u/l), that is, when St equals the bid price.
• Itô’s formula and the ODE yield
dw(Yt ) = −(1 − γ)σ 2 w (Yt )w(Yt )dt + σw (Yt )dWt + w (Yt )(dLt − dUt ).
˜ ˜ ˜ ˜
• Integrate ϕt = w(Yt )Xt /St by parts, insert dynamics of w(Yt ), Xt , St ,
dϕt w (Yt )
= d(Lt − Ut ).
ϕt w(Yt )
• Since Lt and Ut only increase (decrease for µ/γσ 2 > 1) on {Yt = 0} and
{Yt = log(u/l)}, claim follows from boundary conditions for w and w .
81. Transaction Costs
Welfare, Liquidity Premium, Trading
Theorem
Trading the risky asset with transaction costs is equivalent to:
• investing all wealth at the (hypothetical) equivalent safe rate
µ2 − λ2
ESR = r +
2γσ 2
• trading a hypothetical asset, at no transaction costs, with same volatility σ,
but expected return decreased by the liquidity premium
LiPr = µ − µ2 − λ2 .
• Optimal to keep risky weight within buy and sell boundaries
(evaluated at buy and sell prices respectively)
µ−λ µ+λ
π− = , π+ = ,
γσ 2 γσ 2